Cryptography has a very long and fascinating history that dates back to the Egyptian days, some four thousand years ago. However, the best recorded early cryptosystem is the Caesar cipher, used by Julius Caesar of the Roman Empire for military use. The secret key of the Caesar cipher is a single whole number n that governs the number of positions by which all letters of a message to be sent are shifted to the right, in a cyclic fashion. This is a symmetric-key encryption system, since the same key, n, is used to decrypt the message by shifting all letters to the left by n positions, again in a cyclic fashion, to receive the message.
Indeed, symmetric-key cryptosystems, based on elementary mathematical operations of permutations, congruence arithmetic, matrix multiplications, iterations, etc., had been the only ones available till the mid 1970's, when Diffie and Hellman introduced public-key cryptography. Both DES (Data Encryption Standard) and AES (Advanced Encryption Standard) are based on symmetric-key encryption/decryption algorithms. In other words, the decryption key is the same as, or can be easily derived from, the encryption key.
There are two general approaches in symmetric-key cryptosystem design, namely block ciphers and stream ciphers, with block ciphers being more popular due to the success of DES. A block cipher breaks up a plaintext message into blocks of fixed lengths and encrypts one block at a time. Substitution ciphers, transposition ciphers, and product ciphers are block ciphers, with the third scheme being a combination of the first two. On the other hand, a stream cipher treats each letter (or word) of a plaintext as a block of length one to reduce error propagation and the need for cache memory. It is based on generation of a key-stream that assigns a ciphertext one letter (or word) at a time.
In contrast to symmetric-key cryptosystems, public-key cryptography is based on asymmetric-key encryption, with a public key for encryption and a private key for decryption. The first practical public-key encryption scheme was introduced by Rivest, Shamir, and Adleman in 1978, and since then there has been a great deal of mathematical research activities in cryptography that engage modem mathematical tools such as Elliptic Curves and Hyperelliptic Curves.
However, even with recent advances in mathematical research, the significant disadvantages of public-key encryption, particularly the need for extraordinarily long keys and the extremely slow throughput rate, cannot be avoided. On other hand, current symmetric-key approaches require either large blocks (in the case of block ciphers) or long key-streams (in the case of stream ciphers), and for both block and stream ciphers to be secure it is recommended to apply the ciphers multiple times, which in turn requires a larger key-space (or key-set). In this regard, it is worthwhile to mention that even the Rijndael algorithm (in AES) processes an encryption operation in ten rounds when the block and key lengths are both 128 bits, and the key is expanded to a much larger key-space, as large as 128 times the number of blocks.
In summary, current symmetric-key cryptosystems are designed for encryption of long messages only, and hence, are not suitable for real-time applications, such as telecommunications, in which latency cannot be tolerated. On the other hand, although public-key cryptosystems are primarily used for encryption of short messages, their disadvantages as discussed above disqualify them as real-time communication encryption tools.
In this regard, it is noted that the frequency content of voice, image, and video data in telecommunication are highly correlated, and the relatively recent mathematical theory and methods of wavelets have proved to provide very powerful algorithms for processing such data. However, time-frequency or time-scale approaches have not been considered for encryption of voice, image, and video data for real-time communication applications in the literature. The present invention is the first to incorporate both time-scale (time-frequency) and encryption schemes for such applications. In particular, this invention introduces the use of encryption keys in the operation of wavelet transforms to provide additional security.